When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars.
There are several kinds of orbital periods for objects around the Sun:
Relation between sidereal and synodic period
Using the abbreviations
During the time S, the Earth moves over an angle of (360°/E)S (assuming a circular orbit) and the planet moves (360°/P)S.
Let us consider the case of an inferior planet, i.e. a planet that will complete one orbit more than Earth before the two return to the same position relative to the Sun.
and using algebra we obtain
For a superior planet one derives likewise:
Generally, knowing the sidereal period of the other planet and the Earth, P and E, the synodic period can easily be derived:
The above formulae are easily understood by considering the angular velocities of the Earth and the object: the object's apparent angular velocity is its true (sidereal) angular velocity minus the Earth's, and the synodic period is then simply a full circle divided by that apparent angular velocity.
Table of synodic periods in the Solar System, relative to Earth:
In the case of a planet's moon, the synodic period usually means the Sun-synodic period. That is to say, the time it takes the moon to run its phases, coming back to the same solar aspect angle for an observer on the planet's surface —the Earth's motion does not affect this value, because an Earth observer is not involved. For example, Deimos' synodic period is 1.2648 days, 0.18% longer than Deimos' sidereal period of 1.2624 d.
Small body orbiting a central body
In astrodynamics the orbital period (in seconds) of a small body orbiting a central body in a circular or elliptical orbit is:
Note that for all ellipses with a given semi-major axis, the orbital period is the same, regardless of eccentricity.
Orbital period as a function of central body's density
For the Earth (and any other spherically symmetric body with the same average density) as central body we get
and for a body of water
T in hours, with R the radius of the body.
Thus, as an alternative for using a very small number like G, the strength of universal gravity can be described using some reference material, like water: the orbital period for an orbit just above the surface of a spherical body of water is 3 hours and 18 minutes. Conversely, this can be used as a kind of "universal" unit of time.
For the Sun as central body we simply get
Two bodies orbiting each other
In celestial mechanics when both orbiting bodies' masses have to be taken into account the orbital period can be calculated as follows:
Note that the orbital period is independent of size: for a scale model it would be the same, when densities are the same (see also Orbit#Scaling in gravity).
In a parabolic or hyperbolic trajectory the motion is not periodic, and the duration of the full trajectory is infinite.
Published - July 2009
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